CHAPT.XV. SUMMARY OF THE KNOWN SYSTEMS OF SIMPLE GROUPS. 307 



By 118 and 133, the simple group #4(4, 2 2 ), which is isomorphic 

 with by 275, has a complete set of 36 conjugate subgroups 

 4(4, 2) holoedrically isomorphic with the symmetric group on 6 letters. 

 By 136, HA(4, 2 2 ) has a complete set of 216 conjugate subgroups 

 LF(%, 2 2 ), holoedrically isomorphic with the alternating group on 

 5 letters. By 270 274, has a subgroup 960 of index 27. 

 The quotient -group -4(4, 3) of the special Abelian group $4(4, 3) 

 is ( 189) holoedrically isomorphic with _FO(5, 3) and therefore with 

 the abstract group 0. By 114, 4(4,3) contains 3 3 (3 2 -l)3 

 substitutions which leave ^ fixed, so that 4(4, 3) contains a sub- 

 group of index 25920 8 3 4 = 40. By 121, 4(4, 3) contains 

 exactly (3 2 -fl)3 2 substitutions conjugate with TI,_I. But the latter 

 is conjugate with T 2 ,_i, the two being identical in the quotient- 

 group 4(4, 3). Hence 4(4, 3) has a subgroup of index 45. Hence 

 the simple group has subgroups of indices 27, 36, 40, 45, 216. By 

 a lengthy analysis 1 ), it has been shown that contains no subgroup 

 of index < 27. The problem of the determination of the 27 straight 

 lines on a general cubic surface has therefore resolvent equations of 

 degrees 27, 36, 40, 45 but none of degree < 27. 



Since is isomorphic with 4(4, 3), our problem is identical 

 with the problem of the trisection of the periods of hyperelliptic 

 functions with four periods. 2 ) 



CHAPTER XV. 



SUMMARY OF THE KNOWN SYSTEMS OF SIMPLE GROUPS. 



287. In the preceding chapters were derived the following systems 

 of simple groups, with the specified restrictions upon the prime 

 number p and the positive integers m and w 8 ): 



where p n > 3 if m = 2, and d is the greatest common divisor of m 

 and p n 1. 



H0(m, p 2n ): [p nm ( r) m ~\p n(m 



where p n > 3 if m = 2, p n > 2 if m = 3, and g is the greatest 

 common divisor of m and p n + 1. 



1) Jordan, Traits', pp. 319 329. 



2) Jordan, pp. 354 369. 



3) The notations were introduced in 108, 119, 148, 194 and end of 209. 



